Question

The coefficient of performance of a Carnot refrigerator working between $30{}^\circ C$ and $0{}^\circ C$ is
A. 10
B. 1
C. 9
D. 0

Answer 1

Hint: We know that a carnot refrigerator is operated between two temperature range. And an external work is done as input. The coefficient of performance can be calculated using the formula. We have given the value of two temperatures and substituting these values in the formula of coefficient of performance will give us the answer.

Formula used:
Coefficient of performance of refrigerator,
$\beta=\frac{T_{2}}{T_{1}-T_{2}}$
Where temperature used is in kelvin scale.

Complete answer:
Refrigerators are reverse Carnot engines. It is also an example of a heat pump. It takes heat from the cold reservoir and does some work and some amount of heat is transferred to the heat reservoir. The refrigerator mainly has a hot reservoir and a cold reservoir. The coefficient of performance of a refrigerator can be defined as the ratio of heat removed from the cold reservoir to work done to remove that heat. The coefficient of performance depends on outside temperature.

Here we have all the values, just substitute the values we get to find the coefficient of performance of the refrigerator.
Temperature, ${T_1}=30{}^\circ C$=273+30=303K
Temperature, ${T_2}=0{}^\circ C$=273K

By using the above formula, we get coefficient of performance as:
$\beta=\frac{273}{303-273}$

On solving we get coefficient of performance as:
$\beta =9.1$

9.1 can be approximated to 9. Therefore, the answer is option (C).

Note: When we solve this kind of problem, you should remember to convert the unit to SI unit otherwise we will not get the correct answer. There are many formulas used to find the coefficient of performance of a refrigerator. Here as temperature is given, the equation having temperature is used. If the amount of heat absorbed and released was given we could use another formula for finding coefficient of performance.